Centroids

1. Centroids

2. Centroid Principles Object�s center of gravity or center of mass Graphically labeled as

3. Centroid Principles Point of applied force caused by acceleration due to gravity Object is in state of equilibrium if balanced along its centroid

4. Centroid Principles What is an object�s centroid location used for in statics?

5. Centroid Principles One can determine a centroid location by utilizing the cross-section view of a three-dimensional object.

6. Centroid Location Symmetrical Objects Centroid location is determined by an object�s line of symmetry.

7. Centroid Location

8. Centroid Location Since a triangle does not have an axis of symmetry, the location of its centroid must be calculated. First, calculate the area of the triangle. The x component of the centroid, lower case x bar, is the length of the base divided by 3. The y component of the centroid, lower case y bar, is the length of the height divided by 3. In other words, the centroid is located one-third of the distance from the large end of the triangle (or two-thirds of the distance from the pointed end).Since a triangle does not have an axis of symmetry, the location of its centroid must be calculated. First, calculate the area of the triangle. The x component of the centroid, lower case x bar, is the length of the base divided by 3. The y component of the centroid, lower case y bar, is the length of the height divided by 3. In other words, the centroid is located one-third of the distance from the large end of the triangle (or two-thirds of the distance from the pointed end).

9. Centroid Location

10. Centroid Location Equations Complex Shapes

11. Centroid Location Complex Shapes 1. Divide the shape into simple shapes.

12. Centroid Location Complex Shapes Review: Calculating area of simple shapes

13. Centroid Location Complex Shapes 3. Calculate the area of each simple shape. Assume measurements have 3 digits.

14. Centroid Location Complex Shapes 4. Determine the centroid of each simple shape.

15. Centroid Location Complex Shapes 5. Determine the distance from each simple shape�s centroid to the reference axis (x and y).

16. Centroid Location Complex Shapes 6. Multiply each simple shape�s area by its distance from centroid to reference axis.

17. Centroid Location Complex Shapes 7. Sum the products of each simple shape�s area and their distances from the centroid to the reference axis.

18. Centroid Location Complex Shapes 8. Sum the individual simple shape�s area to determine total shape area.

19. Centroid Location Complex Shapes 9. Divide the summed product of areas and distances by the summed object total area.

20. Centroid Location Equations Complex Shapes

21. Common Structural Elements Most structural elements have geometry that can be divided into simple shapes. Each of these shapes can be considered to be part of a rectangular solid. The shapes above include rectangle, L, C, Box, and I.Most structural elements have geometry that can be divided into simple shapes. Each of these shapes can be considered to be part of a rectangular solid. The shapes above include rectangle, L, C, Box, and I.

22. Angle Shape (L-Shape) An angle shape ( L-Shape) can be thought of as being subdivided into two rectangles. This can be done in two different ways.An angle shape ( L-Shape) can be thought of as being subdivided into two rectangles. This can be done in two different ways.

23. Channel Shape (C-Shape) A channel shape (C-Shape) can be subdivided into 3 rectangles. It can be subdivided in four different ways.A channel shape (C-Shape) can be subdivided into 3 rectangles. It can be subdivided in four different ways.

24. Box Shape The box shape can be divided into four rectangles. The box shape can be divided into four rectangles.

25. I-Beam An I-Beam can be divided into three rectangles.An I-Beam can be divided into three rectangles.

26. Centroid of Structural Member In order to calculate the strength of a structural member, its centroid must first be calculated. If the member is subject to pure bending and its stresses remain in the elastic range, then the neutral axis, when seen in two dimensions, or the neutral plane in 3D, of the member passes through the centroid of the section. The centroid is an ordered pair, an x and y position. The centroid is always located on an axis of symmetry. For a simple rectangle (which has two axes of symmetry), the location of the centroid is at the intersection of the two axes of symmetry. Note the symbol used to indicate the centroid.In order to calculate the strength of a structural member, its centroid must first be calculated. If the member is subject to pure bending and its stresses remain in the elastic range, then the neutral axis, when seen in two dimensions, or the neutral plane in 3D, of the member passes through the centroid of the section. The centroid is an ordered pair, an x and y position. The centroid is always located on an axis of symmetry. For a simple rectangle (which has two axes of symmetry), the location of the centroid is at the intersection of the two axes of symmetry. Note the symbol used to indicate the centroid.

27. Neutral Plane The neutral plane is the unstressed region that passes through a structural member and separates the section under tension from the part of the structural member under compression. This can easily be imagined or demonstrated using a loaf of bread that has not been sliced to watch the effects as it undergoes bending.The neutral plane is the unstressed region that passes through a structural member and separates the section under tension from the part of the structural member under compression. This can easily be imagined or demonstrated using a loaf of bread that has not been sliced to watch the effects as it undergoes bending.